mlsanityData to 7 Oct 2026

Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains

arXiv 2104.10873 ↗First posted 22 Apr 2021cs.LG cs.PF physics.comp-phcited by 37

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Abstract (arXiv)

Physics-informed neural networks (PINNs) are increasingly employed to replace/augment traditional numerical methods in solving partial differential equations (PDEs). While state-of-the-art PINNs have many attractive features, they approximate a specific realization of a PDE system and hence are problem-specific. That is, the model needs to be re-trained each time the boundary conditions (BCs) and domain shape/size change. This limitation prohibits the application of PINNs to realistic or large-scale engineering problems especially since the costs and efforts associated with their training are considerable. We introduce a transferable framework for solving boundary value problems (BVPs) via deep neural networks which can be trained once and used forever for various unseen domains and BCs. We first introduce genomic flow network(GFNet), a neural network that can infer the solution of a BVP across arbitrary BCson a small square domain called genome. Then, we proposed mosaic flow(MF) predictor, a novel iterative algorithm that assembles the GFNet's inferences for BVPs on large domains with unseen sizes/shapes and BCs while preserving the spatial regularity of the solution. We demonstrate that our framework can estimate the solution of Laplace and Navier-Stokes equations in domains of unseen shapes and BCs that are, respectively, 1200 and 12 times larger than the training domains. Since our framework eliminates the need to re-train models for unseen domains and BCs, it demonstrates up to 3 orders-of-magnitude speedups compared to the state-of-the-art.

Key facts. Cited by 37 papers in the map (data through 7 Oct 2026). Builds on 34 of its 95 references that are in the map. First posted on arXiv 22 Apr 2021.

Builds on

34 of 95 references.

References · 34

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  1. 2015LSTM: A Search Space Odysseycited by 315
  2. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 7; 2023: 9; 2024: 9; 2025: 10; 2026: 2.

37 papers.

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  1. 2023Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behaviorcited by 128
  2. 2023In-Context Operator Learning with Data Prompts for Differential Equation Problemscited by 98
  3. 2023Multiple Physics Pretraining for Physical Surrogate Modelscited by 90
  4. 2022Deep transfer operator learning for partial differential equations under conditional shiftcited by 83
  5. 2022Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal datacited by 37
  6. 2024PDE Generalization of In-Context Operator Networks: A Study on 1D Scalar Nonlinear Conservation Lawscited by 30
  7. 2025PDE-Transformer: Efficient and Versatile Transformers for Physics Simulationscited by 29
  8. 2025Walrus: A Cross-Domain Foundation Model for Continuum Dynamicscited by 26
  9. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  10. 2024The Challenges of the Nonlinear Regime for Physics-Informed Neural Networkscited by 21
  11. 2024Operator Learning with Gaussian Processescited by 18
  12. 2024Diffeomorphism Neural Operator for various domains and parameters of partial differential equationscited by 15
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  1. 2023BubbleML: A Multi-Physics Dataset and Benchmarks for Machine Learningcited by 14
  2. 2023Machine learning and domain decomposition methods -- a surveycited by 14
  3. 2022Multi-fidelity wavelet neural operator with application to uncertainty quantificationcited by 12
  4. 2024Masked Autoencoders are PDE Learnerscited by 11
  5. 2023On the Generalization of PINNs outside the training domain and the Hyperparameters influencing itcited by 10
  6. 2023Domain decomposition-based coupling of physics-informed neural networks via the Schwarz alternating methodcited by 9
  7. 2024Parametric Encoding with Attention and Convolution Mitigate Spectral Bias of Neural Partial Differential Equation Solverscited by 9
  8. 2024A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equationscited by 8
  9. 2022Local neural operator for solving transient partial differential equations on varied domainscited by 7
  10. 2024The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order modelscited by 4
  11. 2025Hybrid coupling with operator inference and the overlapping Schwarz alternating methodcited by 4
  12. 2022Real-time Health Monitoring of Heat Exchangers using Hypernetworks and PINNscited by 3
  13. 2023A Sequential Meta-Transfer (SMT) Learning to Combat Complexities of Physics-Informed Neural Networks: Application to Composites Autoclave Processingcited by 3
  14. 2023Breaking Boundaries: Distributed Domain Decomposition with Scalable Physics-Informed Neural PDE Solverscited by 3
  15. 2025Residuals Are Not Enough: Limits of Physics-Informed Pre-Training for Scientific Foundation Modelscited by 3
  16. 2025Continuum Transformers Perform In-Context Learning by Operator Gradient Descentcited by 3
  17. 2022A physics and data co-driven surrogate modeling approach for temperature field prediction on irregular geometric domaincited by 2
  18. 2026Tackling Failure Modes of PINNs and PIKANs Using Conflict-Free Gradientscited by 2
  19. 2024Physics-informed Partitioned Coupled Neural Operator for Complex Networkscited by 1
  20. 2025Scale-Consistent Learning for Partial Differential Equationscited by 1
  21. 2025Localized PCA-Net Neural Operators for Scalable Solution Reconstruction of Elliptic PDEscited by 1
  22. 2026Latent Space Element Methodcited by 1
  23. 2025Accelerated Gradient-based Design Optimization Via Differentiable Physics-Informed Neural Operator: A Composites Autoclave Processing Case Studycited by 0
  24. 2025Virtual domain extension for imposing boundary conditions in flow simulation using pre-trained local neural operatorcited by 0
  25. 2025A Flow-rate-conserving CNN-based Domain Decomposition Method for Blood Flow Simulationscited by 0